This post is an expanded version of this MSE post.

Assume that $(M, \omega)$ is a symplectic manifold which is equiped with a Riemannian metric.

Is there a symplectic structure $\omega '$ which is a harmonic $2$-form? Can one choose such a $\omega'$ such that it would be de Rham cohomolgue to the initial form $\omega$?

We consider the following particular case:

For a Riemannian manifold $(M,g)$, we equip the tangent bundle $TM$ with the Sasaki metric $g_s$ and the natural symplectic structure $\omega $ arising from the canonical structure on the cotangent bundle.

Under which conditions on the Riemannian manifold $(M,g)$, is the symplectic $2$-form $\omega$ a harmonic form on $(TM,g_s)$?

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    $\begingroup$ This is always true on a Kahler manifold: see eg here (the comment on the first answer is also relevant). $\endgroup$ – Mike Miller Sep 21 '18 at 19:27
  • $\begingroup$ @MikeMiller thanks for your great comments. $\endgroup$ – Ali Taghavi Sep 22 '18 at 12:06
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    $\begingroup$ Concerning your first question, whether on a symplectic manifold with given metric $g$ there always exists a harmonic symplectic form, I believe the answer should be negative. Though I don't know how to construct an example and don't know a reference. However, if you read what is written in 0.2.B' of Gromov's famous article ihes.fr/~gromov/wp-content/uploads/2018/08/945.pdf , it looks like he doesn't rule out the possibility that for some metrics on $\mathbb CP^2$ the harmonic two-form (unique up to scale) will vanish somewhere. $\endgroup$ – Dmitri Panov Dec 5 '18 at 22:49

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